A sharp L-p-regularity result for second-order stochastic partial differential equations with unbounded and fully degenerate leading coefficients

초록

We present existence, uniqueness, and sharp regularity results of solution to the stochastic partial differ-ential equation (SPDE)du = (aij(& omega;,t)uxixj + f )dt + (& sigma;ik (& omega;,t)uxi + gk)dwkt , u(0, x) = u0, (0.1) where {wkt : k = 1, 2, & BULL; & BULL; & BULL; } is a sequence of independent Brownian motions. The coefficients are merely measurable in (& omega;, t) and can be unbounded and fully degenerate, that is, coefficients aij, & sigma;ik merely satisfy ⎛⎞ () ⠆& INFIN; & alpha;ij(& omega;,t) dxd := ⎝aij(& omega;,t) - 1 & sigma;ik(& omega;,t)& sigma;jk(& omega;,t)⎠ > 0. (0.2) 2 k=1 In this article, we prove that there exists a unique solution u to (0.1), and ( Hux x HH & gamma;p(& tau;,& delta;) & LE; N(d,p) Hu0HB & gamma;+2(1-1/p) +Hf HH & gamma;p(& tau;,& delta;1-p) p �+ ⠅gx ⠅p H & gamma; p (& tau;,|& sigma; |p & delta;1-p,l2) + ⠅gx ⠅H & gamma; p (& tau;,& delta;1-p/2,l2) , (0.3) where p & GE; 2, & gamma; & ISIN; R, & tau; is an arbitrary stopping time, & delta;(& omega;, t) is the smallest eigenvalue of & alpha;ij(& omega;, t), H & gamma;p(& tau;, & delta;) is a weighted stochastic Sobolev space, and B & gamma; +2(1-1/p) p is a stochastic Besov space.& COPY; 2023 Elsevier Inc. All rights reserved.

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